How many permutations of the English alphabet do NOT have all five vowels appearing consecutively?
What I have:
Since there are $26$ letters in the alphabet and each letter can be used only once, there are $26!$ arrangements of the letters in the alphabet in a string. There are $21$ arrangements of the vowels surrounded by the consonants:
1){string of $5$ vowels}{string of $21$ consonants}
2) {string of $1$ consonant}{string of $5$ vowels}{string of $20$ consonants}
...
21){string of $21$ consonants}{string of $5$ vowels}
So there are $22(5!)(21!)$ strings of the alphabet in which all $5$ vowels appear consecutively. So there are $26!-22(5!)(21!)$ strings in which the $5$ vowels do not appear consecutively.
(not sure why part of this is showing up in bold font)